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Adventures in Mathematical Reasoning First Edition
The Role of Spatial Reasoning in Mathematical Thought
Research in Mathematics Education
Series editors
Jinfa Cai, Newark, DE, USA
James A. Middleton, Tempe, AZ, USA
This series is designed to produce thematic volumes, allowing researchers to access numerous studies on a theme in a single, peer-reviewed source. Our intent for this series is to publish the latest research in the field in a timely fashion. This design is particularly geared toward highlighting the work of promising graduate students and junior faculty working in conjunction with senior scholars. The audience for this monograph series consists of those in the intersection between researchers and mathematics education leaders—people who need the highest quality research, methodological rigor, and potentially transformative implications ready at hand to help them make decisions regarding the improvement of teaching, learning, policy, and practice. With this vision, our mission of this book series is:
1. To support the sharing of critical research findings among members of the mathematics education community;
2. To support graduate students and junior faculty and induct them into the research community by pairing them with senior faculty in the production of the highest quality peer-reviewed research papers; and
3. To support the usefulness and widespread adoption of research-based innovation.
More information about this series at http://www.springer.com/series/13030
Kelly S. S. Mix • Michael T. Battista Editors
Visualizing Mathematics
The Role of Spatial Reasoning in Mathematical Thought
Editors
Kelly S. S. Mix
Department of Human Development and Quantitative Methodology University of Maryland College Park, MD, USA
Michael T. Battista
Department of Teaching and Learning
The Ohio State University Columbus, OH, USA
ISSN 2570-4729
Research in Mathematics Education
ISSN 2570-4737 (electronic)
ISBN 978-3-319-98766-8 ISBN 978-3-319-98767-5 (eBook)
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Foreword
In 2016, we were approached by series editor, Dr. Jinfa Cai, with a novel idea— invite authors from the fields of developmental psychology and mathematics education to write about their work on spatial visualization and mathematics, and then ask them to write commentaries on one another’s chapters. The goal was to provide a unique view of research on this topic that encompassed both disciplines, as well as foster cross-field communication and intellectual synergy. We eagerly took up the challenge and invited scholars whose work we knew to be at the forefront of our respective fields. The chapters and commentaries contained in this volume are the products of this esteemed group. They reflect the state of the art in research on spatial visualization and mathematics from at least two perspectives. They highlight important new contributions, but they also reveal the fault lines between our respective disciplines. The commentaries insightfully point out some of these fault lines, as well as the immense common ground and the potential for deeper collaboration in the future.
The basic question of how spatial skill relates to mathematics has received steady attention over the years. In psychology, most of this work has focused on long-term outcomes in STEM fields for individuals with more advanced spatial skill (e.g., Wai, Lubinski, & Benbow, 2009), the possibility that spatial deficits contribute to poor mathematics outcomes in children (e.g., Geary, Hoard, Byrd-Craven, Nugent & Numtee, 2007), and the use of materials that physically embody (via spatial relations) abstract mathematics concepts (see Mix, 2010, for a review). Running through these disparate research programs is the shared notion that spatial thinking plays a major role in understanding mathematics, but it has not been addressed head on in psychology until recent years.
In mathematics education, Clements and Battista, in their 1992 research review, address just this issue. They wrote that both Hadamard and Einstein (renown mathematicians) claimed that much of the thinking required in higher mathematics is spatial, and they cited positive correlations between spatial ability and mathematics achievement at all grade levels. However, even in that time period, the relations between spatial thinking and learning nongeometric concepts did not seem straightforward, and there were conflicting findings. For some tasks, having high-spatial
skill seemed to improve performance, whereas in other tasks, processing mathematical information using verbal-logical reasoning enhanced performance compared to students who processed the information visually. Other mathematics education researchers countered that the understanding of some low-spatial students who did well in mathematics was instrumental, whereas high-spatial students’ understanding was more relational, a difference often not captured by classroom or standardized assessments. Clements and Battista concluded that even though there was reason to believe that spatial reasoning is important in students’ learning and use of mathematical concepts—including nongeometric concepts—the role that such reasoning plays in this learning remained elusive.
Possibly because of this elusiveness, interest in the topic waned in mathematics education. However, currently there is intense interest in this general topic in both psychology and mathematics education due to its potential educational benefits (Newcombe, 2010) and the insights into the relations found through extensive and detailed student interviews (Bruce et al., 2017; Davis et al., 2015). The chapters contributed to this volume represent various approaches to advancing this work in education or moving the work in both fields toward educational application.
The developmental psychology chapters tended to focus on the underlying mental representations used to understand mathematics, and the extent to which these representations already involve, or could be improved by spatial processing. Cipora, Schroeder, Soltanlou, and Nuerk provide a detailed analysis of the link between spatial and numerical processing purportedly demonstrated by spatial-numerical association (SNA) or mental number line effects. They conclude that spatial skills provide a crucial tool for understanding mathematics, but this relation may not be realized in the form of a fixed mental number line. Congdon, Vasileyva, Mix, and Levine examine a deep psychological structure that may underlie a range of mathematics topics—namely, the structure involved in identifying and enumerating spatial units of measurement. They argue that mastery of this structure has the potential to support mathematics learning throughout the elementary grades and perhaps head off misconceptions related to fractions, proportions, and conventional later on. Similarly, Jirout and Newcombe focus on another spatial relation with strong ties to mathematics—namely, relative magnitude—outline its potential role in improving instruction on whole number ordering, fractions, and proportions. Casey and Fell discuss the difference between general spatial skill and spatial skill instantiated in specific mathematics problems, concluding that the most effective way to leverage spatial training to improve mathematics outcomes is likely the latter. They highlight a number of instructional techniques from existing curricula that successfully use spatial representations. Finally, Young, Levine, and Mix considered the multidimensional nature of spatial processing and mathematics processing and the inherent complexity involved in identifying possible instructional levers. Following a critique of the existing literature, including recent factor analytic approaches, they conclude with a set of recommendations for improving these approaches and applying what is already known in educational settings.
The mathematics education chapters discuss the spatial processes involved in specific topics in mathematics. Sinclair, Moss, Hawes, and Stephenson examine
how children can learn “through and from drawing,” focusing on spatial processes and concepts in primary school geometry. They argue that drawing is not innate but can be improved, and they illustrate through fine-grained analysis how the potential benefits of geometric drawing can be realized in classrooms. Gutiérrez, Ramírez, Benedicto, Beltrán-Meneu, and Jaime analyze the spatial reasoning of mathematically gifted secondary school students as they worked on a collaborative, communication-intensive, task in which they were shown orthogonal projections of cube buildings along with related verbal information. The authors related the objectives of students’ actions and their visualization processes and students’ solution strategies and cognitive demand. Herbst and Boileau argue that high school geometry instruction can do more than provide names for 3D shapes and formulas for finding surface area and volume. They illustrate, and invite reflection on their design of, a 3D geometry modeling activity in which students write and interpret instructions for how to move pieces of furniture up an L staircase. Lowrie and Logan discuss how the frequency of encountering, and interacting with, information in visual/ graphic format, including on the web, has increased our need for research on the role of spatial reasoning in students’ encoding and decoding of information in mathematics. To this end, they analyze the representational reasoning of students engaged in tasks that permit different types of representations, from diagrams to equations. Battista, Frazee, and Winer describe the spatial processes involved in reasoning about the geometric topics of measurement, shapes, and isometries. They introduce, and use in their analysis, the construct of spatial-numerical linked structuring as the coordinated process in which numerical operations on measurement numbers are linked to spatial structuring of, and operation on, the measured objects in a way that is consistent with properties of numbers and measurement.
As the chapters and commentaries illustrate, there are still fundamental differences between how researchers in psychology and mathematics education view and investigate the fundamental relations between spatial and mathematical reasoning. However, these differences provide fertile ground for exciting new investigations as each field respectively has advanced knowledge in some areas while leaving gaps in others. The commentaries are a starting point for identifying these points of contact and complementarity. We encourage readers to reflect on how the research in the two fields might be further integrated and how to build productive collaborations between the two sets of researchers. We thank all of our authors for taking a first step in this direction.
Michael T. Battista Department of Teaching and Learning
The Ohio State University, Columbus, OH, USA
Kelly S. S. Mix
Department of Human Development and Quantitative Methodology
University of Maryland, College Park, MD, USA
References
Bruce, C., Davis, B., Sinclair, N, McGarvey, L., Hallowell, D., …, Woolcott, G. (2017). Understanding gaps in research networks: using “spatial reasoning” as a window into the importance of networked educational research. Educational Studies in Mathematics, 95(2), 143–161.
Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 420–464). New York: Macmillan.
Davis, B., & the Spatial Reasoning Study Group. (2015). Spatial Reasoning in the Early Years: Principles, assertions, and speculations. New York: Routledge.
Geary, D. C., Hoard, M. K., Byrd-Craven, J., Nugent, L., & Numtee, C. (2007). Cognitive mechanisms underlying achievement deficits in children with mathematical learning disability. Child Development, 78(4), 1343–1359.
Mix, K. S. (2010). Spatial tools for mathematical thought. In K.S. Mix, L. B. Smith, & M. Gasser (Eds.), The spatial foundations of language and thought (pp. 41–66). New York: Oxford University Press.
Newcombe, N. S. (2010). Picture this: Increasing math and science learning by improving spatial thinking. American Educator, 34(2), 29–43.
Wai, J., Lubinski, D., & Benbow, C. P. (2009). Spatial ability for STEM domains: Aligning over 50 years of cumulative psychological knowledge solidifies its importance. Journal of Educational Psychology, 101(4), 817–835.
Part I Psychological Perspectives
1 How Much as Compared to What: Relative Magnitude as a Key Idea in Mathematics Cognition .
Jamie Jirout and Nora S. Newcombe
3
2 From Intuitive Spatial Measurement to Understanding of Units . . . . 25
Eliza L. Congdon, Marina Vasilyeva, Kelly S. S. Mix, and Susan C. Levine
3 Spatial Reasoning: A Critical Problem-Solving Tool in Children’s Mathematics Strategy Tool-Kit
Beth M. Casey and Harriet Fell
4 More Space, Better Mathematics: Is Space a Powerful Tool or a Cornerstone for Understanding Arithmetic?
Krzysztof Cipora, Philipp Alexander Schroeder, Mojtaba Soltanlou, and Hans-Christoph Nuerk
5 What Processes Underlie the Relation Between Spatial Skill and Mathematics?
Christopher Young, Susan C. Levine, and Kelly S. S. Mix
Part II Commentaries
6 Part I Commentary 1: Deepening the Analysis of Students’ Reasoning About Length
Michael T. Battista, Leah M. Frazee, and Michael L. Winer
7 Part I Commentary 2: Visualization in School Mathematics Analyzed from Two Points of View
A. Gutiérrez
8 Part I Commentary 3: Proposing a Pedagogical Framework for the Teaching and Learning of Spatial Skills: A Commentary on Three Chapters
Tom Lowrie and Tracy Logan
9 Part I Commentary 4: Turning to Temporality in Research on Spatial Reasoning.
Nathalie Sinclair
10 Analyzing the Relation Between Spatial and Geometric Reasoning for Elementary and Middle School Students .
Michael T. Battista, Leah M. Frazee, and Michael L. Winer
11 Learning Through and from Drawing in Early Years Geometry . . . . 229
Nathalie Sinclair, Joan Moss, Zachary Hawes, and Carol Stephenson
12 The Interaction Between Spatial Reasoning Constructs and Mathematics Understandings in Elementary Classrooms . .
Tom Lowrie and Tracy Logan
13 Geometric Modeling of Mesospace Objects: A Task, its Didactical Variables, and the Mathematics at Stake
Patricio Herbst and Nicolas Boileau
14 Visualization Abilities and Complexity of Reasoning in Mathematically Gifted Students’ Collaborative Solutions to a Visualization Task: A Networked Analysis
A. Gutiérrez, R. Ramírez, C. Benedicto, M. J. Beltrán-Meneu, and A. Jaime
Part IV Commentaries
15 Part II Commentary 1: Mathematics Educators’ Perspectives on Spatial Visualization and Mathematical Reasoning
Beth M. Casey
16 Part II Commentary 2: Disparities and Opportunities: Plotting a New Course for Research on Spatial Visualization and Mathematics
Kelly S. S. Mix and Susan C. Levine
17 Part II Commentary 3: Linking Spatial and Mathematical Thinking: The Search for Mechanism
Nora S. Newcombe
18 On the Multitude of Mathematics Skills: Spatial-Numerical Associations and Geometry Skill?
Krzysztof Cipora, Philipp A. Schroeder, and Hans-Christoph Nuerk
Index
Contributors
Michael T. Battista Department of Teaching and Learning, The Ohio State University, Columbus, OH, USA
M. J. Beltrán-Meneu Departamento de Educación, Jaume I University, Castellón, Spain
C. Benedicto Departamento de Didáctica de la Matemática, University of Valencia, Valencia, Spain
Nicolas Boileau Educational Studies Program, University of Michigan, Ann Arbor, MI, USA
Beth M. Casey Lynch School of Education, Boston College, Boston, MA, USA
Krzysztof Cipora Department of Psychology, University of Tuebingen, Tuebingen, Germany
LEAD Graduate School and Research Network, University of Tuebingen, Tuebingen, Germany
Eliza L. Congdon Department of Psychology, Bucknell University, Lewisburg, PA, USA
Harriet Fell College of Computer and Information Science, Northeastern University, Boston, MA, USA
Leah M. Frazee Department of Mathematical Sciences, Central Connecticut State University, New Britain, CT, USA
A. Gutiérrez Departamento de Didáctica de la Matemática, University of Valencia, Valencia, Spain
Zachary Hawes Ontario Institute for Studies in Education/University of Toronto, Toronto, ON, Canada
Patricio Herbst Educational Studies Program, University of Michigan, Ann Arbor, MI, USA
A. Jaime Departamento de Didáctica de la Matemática, University of Valencia, Valencia, Spain
Jamie Jirout Curry School of Education, University of Virginia, Charlottesville, VA, USA
Susan C. Levine Departments of Psychology, and Comparative Human Development and Committee on Education, University of Chicago, Chicago, IL, USA
Tracy Logan Faculty of Education, University of Canberra, Bruce, ACT, Australia
Tom Lowrie Faculty of Education, University of Canberra, Bruce, ACT, Australia
Kelly S. S. Mix Department of Human Development and Quantitative Methodology, University of Maryland, College Park, MD, USA
Joan Moss Ontario Institute for Studies in Education/University of Toronto, Toronto, ON, Canada
Nora S. Newcombe Department of Psychology, Temple University, Philadelphia, PA, USA
Hans-Christoph Nuerk Department of Psychology, University of Tuebingen, Tuebingen, Germany
LEAD Graduate School and Research Network, University of Tuebingen, Tuebingen, Germany
R. Ramírez Departamento de Didáctica de la Matemática, University of Granada, Granada, Spain
Philipp A. Schroeder Department of Psychology and Psychiatry, University of Tuebingen, Tuebingen, Germany
Department of Psychiatry and Psychotherapy, University of Tuebingen, Tuebingen, Germany
Nathalie Sinclair Faculty of Education, Simon Fraser University, Burnaby, BC, Canada
Mojtaba Soltanlou Department of Psychology, University of Tuebingen, Tuebingen, Germany
LEAD Graduate School and Research Network, University of Tuebingen, Tuebingen, Germany
Carol Stephenson Ontario Institute for Studies in Education/University of Toronto, Toronto, ON, Canada
Marina Vasilyeva Lynch School of Education, Boston College, Chestnut Hill, MA, USA
Contributors
Michael L. Winer Department of Mathematics, Eastern Washington University, Cheney, WA, USA
Christopher Young Consortium on School Research, University of Chicago, Chicago, IL, USA
Part I
Psychological Perspectives
Chapter 1 How Much as Compared to What: Relative Magnitude as a Key Idea in Mathematics Cognition
Jamie Jirout and Nora S. Newcombe
Abstract Most topics beyond basic arithmetic require relative magnitude reasoning. This chapter describes the link between relative magnitude reasoning and spatial scaling, a specific type of spatial thinking. We discuss use of the number line, proportional reasoning, and fractions. Consideration of the relational reasoning involved in mathematics can advance our understanding of its relation to spatial skills, and has implications for mathematics instruction, such as using spatial reasoning interventions in developing effective methods for supporting relative magnitude understanding. We review evidence that interventions can be successful in promoting better relative magnitude understanding and associated spatial-relational reasoning, and suggest that education considers ways of including relative magnitude learning, along with more traditional whole-number operations, in early educational efforts.
Is “½” big or small? Reasoning about this question demonstrates two types of numerical reasoning: if you answered that 1/2 is small, because it is less than one, you are reasoning about the number as an absolute value. If your answer is to ask, ½ of what, you are reasoning about relative magnitude. In this chapter, we suggest that number is often interpreted in an absolute sense in mathematics education, though we describe how most topics beyond basic arithmetic require relative
J. Jirout (*)
Curry School of Education, University of Virginia, Charlottesville, VA, USA
e-mail: jirout@virginia.edu
N. S. Newcombe
Department of Psychology, Temple University, Philadelphia, PA, USA
K. S. S. Mix, M. T. Battista (eds.), Visualizing Mathematics, Research in Mathematics Education, https://doi.org/10.1007/978-3-319-98767-5_1
J. Jirout and N. S. Newcombe
magnitude reasoning. We discuss how relative magnitude reasoning might involve a specific type of spatial thinking: spatial scaling. Specifically, use of spatial scaling can contribute to precision in relative magnitude reasoning, perhaps by tapping a more generalized magnitude representation. Focusing on relative magnitude reasoning is an important consideration when determining how spatial thinking relates to mathematics learning, and may have implications for approaches to mathematics instruction, such as using spatial reasoning research and interventions in developing effective methods for supporting relative magnitude understanding. More broadly, consideration of relational reasoning involved in mathematics might advance our understanding of its relation to spatial skills, and even support the inclusion of emphasizing spatial learning as a way to prepare for and support mathematics learning.
The National Research Council has outlined goals for mathematics education in which they suggest that numeracy should be the topic most emphasized early on (NRC, 2009). The content standards of the National Council of Teachers of Mathematics concur (NCTM, 2010). But it can be hard to follow these guidelines because it is not always clear what numeracy or “number” means. In early elementary school, number often means integers, using count words to enumerate sets of discrete objects, and to add and subtract from those sets. In a kindergarten mathematics lesson, for example, students might assemble sets of objects to make a specific number or, later in the year, use sets of objects to do addition and subtraction problems. The kindergarten children are encouraged to think about numbers as referring to these collections, such as telling “how many.” When children think about this kind of number—say, they imagine “5”—they need to recognize the Arabic symbol and recall the word “five,” remember that five comes after four and before six, and, crucially, imagine a set of five objects. They might also know that five can be used to refer to a time or date, or to a five-dollar-bill or a 5-year-old child. But in most cases, they are thinking about the value of a positive integer referring to a collection of discrete objects.
In later grades, lessons would be different. Third graders might be learning how to move a decimal point to convert percentages to proportions. Fifth graders might be learning about remainders in long division problems. These lessons involve a fundamentally different kind of “number” than integers. Older children need to think about numbers as a number system, which can quantify many kinds of referents, including referring to continuous magnitudes that need not, and often do not, denote collections of discrete objects. When comparison of fractions is required, when calculating a proportion or percentage, or when dividing one number into another but with some quantity remaining, relative magnitude is key. How is this shift, from number as discrete objects to the meaning of number, dependent on the specific problem?
To determine how to support children’s underlying representation of number as it becomes more complex, it is important to first make the different representations explicit. Recently, Newcombe and colleagues defined the different categories of quantification types across two dimensions (Newcombe, Frick, & Möhring, 2018). A first distinction is that number may refer to collections of discrete objects, as
Fig. 1.1 Newcombe et al.’s (2018) categorization of systems of quantification, modified 1 How
shown in the bottom row of Fig. 1.1. However, the number system can also be used to quantify continuous magnitudes, as shown in the top row of Fig. 1.1. A second distinction is that numbers may refer to relative (or intensive or proportional) quantities, as shown in the left column of Fig. 1.1, or to absolute (or extensive) quantities, as shown in the right column of Fig. 1.1. Jointly, these two dimensions generate a four-cell classification system. Kindergarten mathematics generally focuses on the bottom right, but eventually, children must learn about the whole system (Common Core State Standards Initiative, 2010). Thus, thinking about numbers in terms of these varied uses and meanings is important in understanding the development of mathematical cognition, and in identifying effective ways of supporting students’ representation of number as they continue in their education.
The distinctions of quantification categories are also important in understanding the link between mathematical and spatial thinking (Newcombe et al., 2018; Newcombe, Levine, & Mix, 2015). Spatial reasoning has multiple aspects, including a distinction between the spatial characteristics of an object itself (intensive) and the spatial position of an object in relation to other objects and its surroundings (extensive) (Newcombe & Shipley, 2015). In this chapter, we focus on the overlap between the relational reasoning processes involved in mathematics and spatial reasoning tasks. We focus on relative magnitude, a concept that is fundamental both to spatial scaling and to proportional reasoning of the kind shown in the left column of Fig. 1.1. Spatial relational reasoning skills are seen in young children and even infants, and could help support similar relational reasoning processes in mathematic tasks involving relative magnitude.
This chapter will begin with examples of relative magnitude in mathematics education, examining number line understanding, proportional reasoning, and fraction learning. We then provide a review of research on relative-magnitude reasoning in
J. Jirout and N. S. Newcombe
spatial processes, especially spatial scaling, and how these spatial processes relate to mathematic skills and are utilized when using external representations. We return to number line, proportional reasoning, and fraction tasks to discuss how spatial representations, and thus spatial-relational reasoning, are used in learning. Finally, we conclude with a discussion of interventions shown to improve children’s relative magnitude understanding, and possible connections between spatial learning and developing mathematic skills. We end with a discussion of potential implications of the research on spatial thinking and relative magnitude for mathematics education.
Relative Magnitude in Mathematics Learning
The importance of relative magnitude is evidenced by its necessity for many mathematics tasks both in education and in everyday life. Simple questions like whether a number is “big” or “small” must be evaluated relative to some comparison or scale, for example knowing where to put the number nine on a number line depends on the range of the line (i.e., toward the end on a 0–10 line, but toward the beginning on a 0–100 line). Relative magnitude is important in thinking about proportions to determine what things are being compared and how, for example knowing how much sugar to add for a cup of lemonade when using one lemon, if you know you use a whole cup of sugar for a pitcher using four lemons. Early fraction tasks such as dividing something among friends require understanding the meaning of a partwhole relation, for example, that as the number of friends sharing a cake (the denominator) becomes larger, the portion of the cake that each receives becomes smaller. These different tasks share the common cognitive process of relative magnitude reasoning. We explain this idea further for each of three mathematical tasks now, and return to these tasks later in the chapter to discuss relations across relative magnitude tasks and how spatial representations influence relative magnitude reasoning in the tasks.
Number Lines as Relative Magnitude
Although some mathematic tasks in research measure absolute value knowledge, others, including the widely used number line estimation, involve relative magnitude reasoning. In early development, understanding of number is often assessed with tasks asking children to give a specified number of objects, or the “give-N” task (e.g., Wynn, 1990). But many studies of mathematics cognition or number understanding use measures of relative rather than absolute magnitude reasoning (e.g., using estimates of more or less rather than absolute value). Rather than asking how many, these tasks rely on children considering how much/many compared to what. For example, on the widely used number-line estimation task, children are asked to show “how much” of the given line is equivalent to a specific value, but often they
Fig. 1.2 Example of the number line estimation task. In one version, the participant is asked to mark a line to show where a given Arabic numeral would go, relative to the scale provided. In another version, the participant is asked to give the value of the mark provided on the number line, relative to the scale provided. Here, the mark shows a value of 50 relative to the 0–100 scale. If the scale changed to 0–1000, the mark would show 500 1 How Much as Compared
are provided beginning and endpoint values, including a scale for comparison to the value (Siegler & Booth, 2004). The child’s specific task is to place a mark on the line to represent where a specific number would fall, or to provide an estimate of the number represented by a mark shown on the line (see Fig. 1.2). As opposed to the give N tasks, successful performance on the number line task requires relative magnitude understanding. The magnitude of the space between the provided endpoints is relative to the scale (i.e., 1 in. on a 10-in. number line represents a single unit for a scale of 0–10, but represents 10 units if the scale is 0–100). This requires children to shift from their early mathematics experience in thinking about number as absolute values of discrete objects, to thinking about number as a value relative to the scale. Younger children sometimes use a more familiar discrete counting strategy, ignoring the provided endpoint value. This can result in overestimating lower numbers across the line, and then squeezing the larger numbers toward the end of the scale, resulting in a logarithmic representation of the number scale. As they get older, children begin to use more proportional strategies and thus their representations become more linear, though a shift back to the earlier strategy is observed as the scale increases (e.g., 0–1000 to 0–10,000). That is, children progress from basic concepts such as knowing that the word two means two objects, to having linear representations of numbers on scales of 1–10 (age 3–5), 1–100 (age 5–7), 1–1000 (age 7–11), and eventually understanding fractions in a similar way, beginning around age eight, advancing through adulthood (Siegler & Braithwaite, 2017).
In mathematics education, much emphasis is placed on understanding the absolute magnitude of numbers in early elementary school. Yet, even as early as first grade when addition and subtraction are emphasized, current standards explicitly mention the importance of understanding relative magnitude of numbers as well (Common Core State Standards Initiative, 2010). As children begin to learn more complex mathematics, reasoning about relative magnitude becomes much more central, as the content begins to rely much more on relational reasoning than absolute values of number when fractions, proportions, functions, probabilities, etc. are introduced (Common Core State Standards Initiative, 2010). Beginning in third grade, children whose education thus far focused on the absolute value of numbers are expected to shift to reasoning about relative magnitude, where paying attention to whole numbers would lead to less accurate performance (DeWolf & Vosniadou, 2011). It is important, then, to consider how this shift can be supported in educational practice.
J. Jirout and N. S. Newcombe
Proportional Reasoning
Proportional reasoning is typically used to solve a problem in order to reach a specific value of interest, yet, like number line estimation, it requires relative magnitude reasoning. Reasoning about proportion is observed in everyday problems, such as when comparing costs of two items that are different prices and amounts or to estimate total cost with sales tax. In mathematics instruction, proportional reasoning is necessary when calculating concentrations of a solution, or in traditional multiplication problems such as using a given proportion—say, the speed a car travels—to determine how long it will take for the car to reach a destination of a specific distance. This task is dependent on the ability to reason about one quantity relative to another, determining a ratio, and often to apply this relational information to another context. Though the algorithmic use of formulas often studied to solve this type of problem does not seem to involve relative magnitude (i.e., plugging numbers in), many direct measures of proportional reasoning require children to use relative magnitude reasoning. For instance, proportional reasoning tasks in research typically use spatial displays of concentrations, such as proportions of water to juice, with children choosing a matching concentration that would taste the same (i.e., have the same concentration) or showing the concentration using a scale from very weak to very strong (Boyer, Levine, & Huttenlocher, 2008; Möhring, Newcombe, Levine, & Frick, 2016a).
Fraction Learning
Like proportions, fractions are part-whole relations in which their meaning is derived from relative magnitude. But unlike proportions, fractions are considered numbers themselves that can be represented on a number line, and they can be greater than one. In fact, many researchers and educators argue that thinking about fractions as numbers by representing them on a number line (relative to other fractions and whole numbers) improves learning. Specifically, the widely used Common Core standards suggest that students learn to place fractions on number lines, implicitly considering them to be absolute magnitudes (Common Core State Standards Initiative, 2010). Yet when fractions are included in mathematics problems, understanding them as relative magnitudes helps conceptual understanding. For example, it is fairly easy to place the fraction one half between zero and one on a number line, but if you are multiplying a number by one half, it is helpful to think of it as signifying one of two equal parts of a quantity, which may of course be much greater than one. The relation between the numerator and denominator is what makes a fraction meaningful, and knowing that this relation can be applied to any quantity just as proportions can be. Understanding fractions is important for success in algebra (NMAP, 2008), with algebra considered to be “the gatekeeper to higher learning in mathematics and science” (Booth & Newton, 2012, p. 247), and it lays
1 How Much as Compared to What: Relative Magnitude as a Key Idea
the foundation to more advanced mathematics. It has been suggested that the relation between fraction knowledge and later success in algebra may be due to more general underlying knowledge of number systems and magnitude, supporting more abstract mathematical reasoning (Ketterlin-Geller, Gifford, & Perry, 2015).
Relative Magnitude and Spatial Thinking
Similar to the varied conceptualizations of number, spatial thinking is a broad label for many different types of reasoning. For example, planning how to navigate from one point to another requires spatial thinking, but so does determining which size pot will hold the amount of food you plan to cook; children’s spatial thinking in play can involve matching up pieces of a puzzle or building with Lego diagrams, or comparing lengths of sticks to choose which should be the daddy vs. baby. Spatial thinking is also multidimensional, similar to the conceptualizations of number.
Relational thinking can be observed in intensive spatial tasks similar to number tasks. For example, relational position of discrete objects in a model or diagram (i.e., attending to the relative position within an array) is used to solve spatial analogies. In arrays of toys where array 1 is “pig,” “dog,” “chick” and array 2 is “horse,” “mouse,” “dog,” the relational match to “dog” from array 1 is “mouse” in array 2. This relational matching of relative position can also be done with continuous spaces, such as finding a location in real space on a map. When using a map, the task draws on relative magnitude when determining the position of a target in the continuous space, creating a representation or estimation of relative position that can then be applied to another space. When this relational matching is done across spaces and/or representations of different size, the process of spatial scaling is used. Spatial scaling is one area of focus in research investigating the relation between spatial thinking and mathematics tasks requiring relative magnitude.
Spatial Scaling
Spatial scaling is the ability to reason about spatial relations in one context and to apply this relational information to a different sized spatial area. Just as proportions and fractions often involve identifying relational information and applying it to solve a problem, spatial scaling involves two steps: first, recognizing the relational correspondence between the two areas; second, mentally transforming the spatialrelational information from one space to the other (Möhring, Newcombe, & Frick, 2014). An example of spatial scaling is the process used when looking at the distance between two points on a map and then determining that distance in real space. Just as mathematics-relational reasoning can be assessed with number-line estimation, spatial-relational reasoning can be assessed using a spatial scaling measure (see Fig. 1.3). Instead of a symbolic number, children are shown a scaled map
J. Jirout and N. S. Newcombe
1.3 Example of a spatial scaling task with a map (right) and referent space
1.4 Examples of diagrams requiring relational-reasoning, shown for every-day tasks (left), mathematics learning (middle), and science learning with microscopic (right-top) and telescopic (right-bottom) scales
marking a location, and are asked to match that location on a larger space (Frick & Newcombe, 2012).
By definition, spatial scaling requires a representation and a referent. A key function of representations is to convey various types of relational information about the referent. Examples of everyday representation use include navigating a space (referent) that is larger than one can see from a single viewpoint, such as with scaled maps (representation), or showing things too large or small to be seen with the human eye (referent), like in scaled photographs (representation). Other examples include configuring complex materials into a desired state, like when using building instructions, or for displaying information to show relations visually as charts and graphs do. In all of these examples, relational reasoning is critical, and scaling is often used—both in understanding the scale of the representation and, in several cases, extrapolating the information to apply it to another scale (e.g., navigating through a park, or determining which correct size part to use for a step when building furniture, see Fig. 1.4).
Focus on superficial or perceptual information can lead to ineffective use of the representation, for example not going far enough when driving because the distance on a map looked small, while the scale actually indicated it was quite far. Similarly, focus on perceptual rather than relational information in mathematics learning can cause difficulty. For example, in the original conservation of number tasks, Piaget found that children focused on the perceptually salient spatial information of how far a sequence of objects stretched as indicating that it had more, even after just
Fig.
(left)
Fig.
observing that the sequence was equivalent to another before being rearranged (Piaget, 1952). In more advanced magnitude comparison, children often struggle with ordering fractions by magnitude because although magnitude decreases as the denominator increases, they mistakenly associate this increase with an increase in magnitude (e.g., saying one fourth is larger than one half, because four is greater than two).
Research on performance in proportional reasoning and fractions shows a correlation between these processes and spatial scaling. Despite these tasks being different, this relation supports the common reliance on relative magnitude for successful performance (Möhring et al., 2016a; Möhring, Newcombe, Levine, & Frick, 2016b). More broadly, spatial magnitude information is used when interpreting numerical magnitude, for better or for worse (Newcombe et al., 2015); spatial thinking is necessary when completing tasks like placing numbers on a number line, but can also cause bias or error such as Piaget’s number conservation mistakes or when estimations are pulled toward discrete markers or benchmarks. Further, children often learn about proportional reasoning and fractions using diagrams and representations, thus it is important for them to understand how to use them correctly, which often requires spatial scaling.
In fact, the importance of understanding scale of diagrams and using scaled diagrams, as well as an emphasis on scaling skills more generally, is made explicit by its inclusion throughout the common core mathematics standards (Common Core State Standards Initiative, 2010). Although early developmental theory suggested that processes like proportional reasoning are well beyond what young children are capable of (e.g., Piaget, 1952; Piaget & Inhelder, 1956), measures of relational reasoning, especially spatial-relational reasoning and magnitude understanding, suggest that even infants show early aptitude and children become progressively more skilled in early years, with both improved scaling accuracy and more advanced proportional reasoning with age (Lourenco & Longo, 2011).
Spatial Representations
Visual representations like diagrams are common in and important for mathematics education (National Council of Teachers of Mathematics, 2010). These representations rely on spatial thinking, but also have the potential to help develop relationalreasoning skills (Davies & Uttal, 2007; Gentner, 1988) and promote broader learning in mathematics (Woodward et al., 2012). Spatial-relational reasoning might be specifically important for fully understanding and using representations. Representations are not always helpful, for example, if they are perceived as pictures or objects of their own rather than as representing information or a referent (Garcia Garcia & Cox, 2010; Uttal & O’Doherty, 2008). Several varying factors of implementing representations might influence their effectiveness for learning: the structure of representation implementation, type of contact with the representations, the relation of the representation to familiar symbols, and the amount of exposure to 1 How Much as Compared
J. Jirout and N. S. Newcombe
the representations (Mix, 2010). The type of representation and the learning context can vary further in the mechanisms of support representations may offer, from reducing fine motor and cognitive demands and to providing metaphors or embodied experiences (Mix, 2010). The implementation methods, support sought and available from representations and needs of different learners, and the specific learning goals more broadly are all important considerations in using representations for mathematics instruction. Importantly, though, children must have the necessary skills for understanding representations in the first place, such as their representational nature (Uttal & O’Doherty, 2008).
As discussed above, interpreting information from spatial representations often requires spatial scaling. Although children rarely receive explicit instruction on scaling, research shows they are capable of spatial scaling from very young ages. Success in using scaled representations relies on matching the relational structure of the representation, which they learn to do quite early (Gentner, 1988). Failure in this step, for example, would be in choosing the absolute rather than relative size match in Fig. 1.4 (left). Children develop more accurate scaling ability in the early elementary school years, though even adults perform poorly when reasoning about very small or very large scales, thus there is room for improvement. One potential mechanism for developing spatial scaling is improved relational reasoning, such as learning the related processes of proportions or fractions taught in late elementary and middle school grades. Alternatively, early practice with scaled spatial representations could help prepare children for later mathematics learning by strengthening their relational reasoning.
To understand how spatial representations might help support learning of relative magnitude mathematics tasks, an important question is, how do children learn relational reasoning with spatial representations? Though instances of children’s relational thinking can be observed in very young children, and even infants, research shows improvement in representation use, and changes in strategies or processes involved, with age. For example, children demonstrate the ability to understand symbolic correspondence, or that a model represents something else, around the age of 3 years old (DeLoache, 1987). However, understanding spatial representations typically also requires geometric correspondence (Newcombe & Huttenlocher, 2000). Children must be able to encode length or distance within representations and then map that to the referent space, while conducting any necessary scaling if the representation is a different size (Möhring et al., 2014). Young children can successfully use models, typically by using landmark mapping and perceptual coding of the space, and there is some evidence they use categorical or even distance coding (Huttenlocher, Newcombe, & Vasilyeva, 1999). However, coding distances and scaling are not simple tasks. Difficulty can easily be increased by requiring children to code distances along both horizontal and vertical axes or by removing landmark cues. Even adults completing a simple scaling task (i.e., using a map to mark a target on a two-dimensional space) show variation in accuracy as a function of the scale of the representation to the referent space (Möhring et al., 2014).
The adult strategy for using scaled representations is likely determined by the need for accuracy; while adults can quickly and quite accurately interpret scaled
information using perceptual methods (on familiar scales), they are also able to use a slower metric coding strategy, which should provide even more precision. This more sophisticated metric strategy is difficult for children, because it relies on proportional reasoning. Though they do not calculate exact proportions, young children have been found to be capable of using basic spatial categories—for example using a midpoint to create categories within the space (Huttenlocher, Newcombe, & Sandberg, 1994). Just as estimation errors on the number line are seen around benchmarking points, suggesting the use of proportional reasoning strategies, spatial estimation responses also show error patterns that suggest they are being slightly biased by these categorical boundaries, where their responses are pulled toward corners or midpoints (Sandberg, Huttenlocher, & Newcombe, 1996; Slusser, Santiago, & Barth, 2013). This more implicit or categorical way of reasoning about proportions might prepare children for later mathematics learning that requires relative magnitude understanding. Further, spatial representations can be used specifically to help children reason about relative magnitude in mathematics learning.
Spatial Representations in Relative Magnitude Mathematics Tasks
We return now to our discussion of the relative magnitude mathematics tasks discussed above, with a focus on spatial representations and how they might support mathematics learning. Common Core standards include using number lines (similar to the research task discussed earlier) to represent whole numbers and fractions (Common Core State Standards Initiative, 2010). Studies show that number line estimation performance relates to spatial thinking, perhaps by emphasizing the spatial importance in number understanding as a linear representation of numbers (Gunderson, Ramirez, Beilock, & Levine, 2012). The question of what specific processes are involved in number line estimation is the subject of much research. Many studies show that there are spatial patterns in developing number line estimation skills by what is described as a representation shift. Performance on the number line estimation task transitions from estimates that are better fit by a logarithmic function (i.e., lower numbers tend to be, with larger numbers being grouped too closely toward the top of the scale) to a linear fit where numbers are placed relatively equally spaced apart and in order. Children’s estimations become more linear with age, and the timing of this transition is positively correlated with scale; the older children get, the higher the scale showing a linear representation (Siegler & Opfer, 2003). This body of research emphasizes this representation shift as an important developmental milestone of number understanding, for good reason, but it should also be noted that the shift is referring to children’s spatial representations. One reason linear estimates on the number line are important is that they show understanding of equal spacing between whole numbers, and correct knowledge of
order in which the numbers follow; this tends to occur when scales are familiar to the participant (Ebersbach, Luwel, Frick, Onghena, & Verschaffel, 2008). Yet even when one knows the order of numbers, the one-to-one correspondence on a number line is not as easy as counting objects—the space provided must be divided into the equal units. Studies investigating how people use the given scale to determine where a number goes provide evidence that relational reasoning is involved. Specifically, studies using eye-tracking and differences in error patterns show that both children and adults use proportional reasoning strategies when making estimations (Barth & Paladino, 2011; Schneider et al., 2008; Slusser et al., 2013). These studies indicate the use of landmarks—the endpoints as well as fractional landmarks (e.g., ½, ¼). The level of proportional benchmarking relates to improved performance; the more precise proportions used (i.e., more benchmarks), the more accurate estimations were (Peeters, Verschaffel, & Luwel, 2017). Peeters et al. (2017) found that providing explicit proportional markers can induce finer grained proportional reasoning with landmarks, but the use of implicit landmarks (i.e., the proportional ones) also appears to increase with age (Slusser et al., 2013). Thus, not only is children’s developing number line estimation measured as a spatial representation, it also draws on mathematical reasoning related to relative magnitude understanding, and can be improved using spatial-relational interventions with representations. The use of proportional landmarks is observed in other mathematics tasks involving representations as well. Children at the earliest ages of formal schooling show the ability to use ½ as a benchmark when comparing proportions (Singer-Freeman & Goswamani, 2001; Spinillo & Bryant, 1991). On basic part-whole fraction tasks, emerging understanding appears by 4 years of age (Mix, Levine, & Huttenlocher, 1999; Singer-Freeman & Goswamani, 2001). Some research on early proportional reasoning suggests that the specific features of diagrams or materials used to represent proportions might impact children’s attention to the relative vs. absolute magnitude information. Boyer and Levine (2012) asked children to match proportions displayed using a “juice task,” where different concentrations of juice to water were displayed using red and blue segments (see Fig. 1.5). Sometimes the diagram included a single red and a single blue bar, differing in size, to show continuous proportion of whole (e.g., 50% each; Fig. 1.5, bottom). Other diagrams showed the
Fig. 1.5 Discrete vs. continuous proportional representations
J. Jirout and N. S. Newcombe
1 How Much as Compared to What: Relative Magnitude
same information, but the blue and red bars were divided using discrete units (e.g., two red, two blue; Fig. 1.5, top). Children were then asked to choose which of two options was a match to the concentration (i.e., “Which of these two would taste like Wally Bear’s Juice?”). Children were better at matching the continuous bars, though their performance decreased as the comparison became less similar by becoming increasingly different in size (for example, choosing 2/8 instead of 4/8 as a match for 2/4). This effect of scale difference is similar to that observed in tasks of spatial scaling, where performance decreases as the scale factor increases (Möhring et al., 2014). When the visualizations included the discrete units, children often defaulted to an inaccurate counting strategy of the units shown rather than using proportional reasoning; when they were asked to produce a matching proportion, the lines of the discrete units actually amplified errors across scale factors, pulling responses further from the correct location compared to the continuous displays (Boyer & Levine, 2012). This difficulty in applying proportional information from one display to another with discrete units seems unfortunate, since real-world use of proportional reasoning often involves discrete units. For example, if you figure out that pancakes come out best when you combined 2-tbs water to 3-tbs boxed mix and want to now make more than a single pancake, you can determine the proportional composition of 2:3 or 40% water to 60% boxed mix, and then apply this proportion to a unit of measurement (rather than estimating), for example 1-cup water to 1.5-cups mix. The key is to translate the initial discrete amounts into a proportion, rather than trying to apply the raw values from the first amounts. It may not be that children are unable to apply proportional information to quantities of discrete units, but that they are being distracted from determining the proportional information in the first place.
Boyer and Levine (2015) tested this by having children select a proportional match with the comparison proportion presented in spatial representations using either discrete or continuous amounts. As expected, when children are presented first with proportions shown as continuous quantities, they then do better on proportions shown as discrete quantities, though only the older children benefited from the different displays (Boyer & Levine, 2015). The authors suggest that providing the continuous representation first encourages more spatial proportional reasoning strategies rather than a “count-and-match” method. More generally, these finding suggest that spatial visualization using representations might help direct attention to relative magnitude (i.e., the continuous presentation) rather than whole number counting (i.e., the discrete presentation).
Spatial representations of proportions like those in Fig. 1.5 can also be used to show fractions concepts, e.g., 1:2 or 50% is the same as ½, and fractions can be similarly displayed using part-whole diagrams. However, fractions can be larger than one to include any rational numbers, sometimes making diagrams of partwhole relations confusing. Reasoning about fractions can also become difficult when the fraction is to be applied to another number, such as multiplying or dividing by a fraction. Even a simple magnitude comparison task can be challenging if the numbers in the larger fraction have smaller components, (e.g., 3/4 vs. 8/20; LortieForgues, Tian, & Siegler, 2015). Siegler and Lortie-Forgues (2017) outline the reasons why fractions and rational numbers more generally are difficult to understand,
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pale as death, but there was a sweet sunshine upon her face, that seemed to cast a gleam of light through the dungeon. She was sitting by your father, and telling him some pleasant tale, for I saw the old man smile—though the place was very dim.”
Alexis wrung his hands and groaned in an agony of impatience— but he still commanded himself so as to allow Linsk to proceed.
“Well—they were delighted to see me; and your sister, taking me apart, told me to go to the princess Lodoiska, and take to her a ring, and tell her that Pultova of Warsaw and his daughter were in prison, and to beg her immediate aid. I went to find the princess immediately, but she was gone to Poland. In the mean time, your father was tried and condemned. In this state of things, Krusenstern, who was in love with your sister, told her that if his love could be returned, he would save her father. She spurned him as if he had been a serpent, and this turned his heart to gall. Now he seems anxious that your father should die, and the fatal day is fixed for a week from to-morrow.”
Alexis seemed for a few moments in a state of mind which threatened to upset his reason: but soon recovering himself, his step became firm, and his countenance decided. “Take me,” said he, “to the prison, Linsk: I want to see my father and sister without delay.” They went to the place, but found that they could not be admitted. What now could be done? “I will go to the emperor,” said Alexis—“I will appeal to him.”
At this moment the message committed to him by count Zinski, came into his mind. Proceeding to the hotel, he made the most rapid preparations in his power, for proceeding to the palace. This, however, was a work of several hours. At last he set out. Dismounting from the carriage at the gate of the palace, he entered, and as he was crossing the court, a coach with a lady was passing by. At this moment, the horses took fright at some object, and rearing fearfully for a moment, set forward at a full run. They swept quite round the circular court with desperate fury, and were now approaching Alexis.
Springing suddenly upon them, he fortunately seized the bridle of one of the horses, and by his vigorous arm, arrested the progress of the furious animals. While he held them, the driver descended from his box, opened the door of the coach, and the lady, almost fainting from fright, sprung forth upon the ground. Alexis now approached the lady, and was about to offer to conduct her across the court to the palace, when some of the servants, who had witnessed the scene we have described, came up, and gave their assistance. As the lady was moving away, she spoke to Alexis, and asked his name. “Alexis Pultova,” said he.
“Pultova? Pultova?” said the lady, “Alexis Pultova, of Warsaw?”
“Once of Warsaw, madam, but now of Tobolsk.”
“Come, young gentleman,” said the lady, seeming at once to have recovered from her fright, “you must come with me.” Accordingly, she took the arm of Alexis, and they entered the palace. After passing through several halls and galleries, they came to a small room, which the lady entered and Alexis followed.
It is unnecessary to give the details of the interview. The lady was the princess Lodoiska, who had just returned from Poland. The story of count Zinski was soon told, as well as that of the father and sister of Alexis. The princess seemed at first overwhelmed with the double calamity which seemed to fall like shocks of thunder upon her ear. She saw at once the danger to which Zinski, whom she still loved with devoted attachment, had exposed himself by his rash return: and she also felt the extreme difficulty of controverting the artful and villanous scheme of the wicked Krusenstern, in respect to Pultova and his daughter.
She begged Alexis to delay his interview with the emperor a single day, and promised her utmost efforts in behalf of all those in whom Alexis felt so deeply interested. When he was gone, she went straight to Nicholas, and told him the story of the count, as she had heard it from Alexis. She then told frankly her feelings, and stated the circumstances of their former acquaintance, which have already been detailed to the reader. She then threw herself upon her knees, and begged for the life and liberty of her lover.
We need not say that it was a touching plea—but the emperor seemed unmoved, and positively refused to grant the request. He insisted that the count’s crime was one of the highest nature, and it was indispensable that he should receive a signal punishment. “His fate is sealed,” said Nicholas, firmly, “and it shall be executed tomorrow. I hope, fair lady, if you do not approve my mercy, you will at least acknowledge my justice.”
Baffled and broken-hearted, the princess left the stern monarch, and sought her room. On the morrow, Zinski was taken to the castle of St. Petersburgh, and preparations for his execution seemed to be immediately set on foot. In vain was the petition of Lodoiska: in vain the representations and the prayers of the captain of the Czarina. When Alexis came, and delivered the message of Zinski, Nicholas seemed to feel a touch of emotion; but it appeared to pass immediately away.
About four o’clock in the afternoon of the day fixed for the count’s execution, there was a heavy sound of musketry in the court of the castle, then a dead silence, and finally a gate was opened, and a coach, briskly drawn, issued forth, wending its way to the palace of the emperor. A man of a noble form, and still youthful, issued from the coach, and was conducted to the audience room of the Czar.
There stood Nicholas—a man of great height, and vast breadth of shoulder, as if he had been made as the very model of strength: at the same time, his countenance, lighted up by a full blue eye, expressed, amid a lofty and somewhat stern look, an aspect almost of gentleness. By his side was the princess Lodoiska.
The stranger entered the hall, and proceeding toward the emperor, was about to kneel. “Nay, count Zinski,” said his majesty, “we will not have that ceremony to-day. You have been shot, and that is enough. I owe you my life, count, and I am glad of being able to testify my gratitude. I sentenced you to Siberia, expecting that you would petition for reprieve; but you were too proud. I have long mourned over your stubbornness. Your return has given me pleasure, though I could have wished that it had been in some other
way I could not overlook your crime, so I ordered you to be shot— but with blank cartridges. And now, count, what can I do for you?”
“One thing, sire, and but one.”
“What is it?—you shall have your wish.”
“The restoration of Pultova and his family.”
“It cannot be—it cannot be! The rebel has just returned from Tobolsk, like yourself.”
“Then, sire, let him be like me—forgiven.”
“You are ready with your wit, count—but you shall have your way I will give immediate orders for the liberation of Pultova; and he, as well as yourself, shall be restored to your estates at Warsaw.”
Shooting wild geese.
Wild Geese.
T passage of wild geese to the north commences with the breaking up of the ice; their first appearance in Canada and on the shores of Hudson’s Bay, varying with the forwardness of their spring, from the middle of April to the latter end of May Their flight is heavy and laborious, but moderately swift, in a straight line when their number is but few, but more frequently in two lines meeting in a point in front. The van is said to be always led by an old gander, in whose wake the others instinctively follow. But should his sagacity fail in discovering the land-marks by which they usually steer, as something happens in foggy weather, the whole flock appear in the greatest distress, and fly about in an irregular manner, making a great clamor. In their flights, they cross indiscriminately over land and water, differing in this respect from several other geese, which prefer making a circuit by water to traversing the land. They also
pass far inland, instead of confining their course to the neighborhood of the sea.
So important is the arrival of geese to the inhabitants of these northern regions, that the month in which they first make their appearance is termed by the Indians the goose moon. In fact, not only the Indians, but the English settlers also depend greatly upon these birds for their subsistence, and many thousands of them are annually killed, a large proportion of which are salted and barrelled for winter consumption. Many too that are killed on their return, after the commencement of the frost, are suffered to freeze, and are thus kept as fresh provisions for several months.
Travels, Adventures and Experiences of Thomas Trotter.
.
The plain of Lombardy—Inundations of the Po—Padua—First prospect of Venice—Arrival at the city—Description of Venice— Character of the Venetians—Contrast of the Italians with the Americans—Journey to Milan and Turin—The plain of Marengo— Genoa—Leghorn—Return to Boston—General observations on travelling in Italy—Conclusion.
T remainder of our journey to Venice was through a region which exhibited a remarkable contrast to that which we had recently traversed. No mountains nor even hills diversified the face of the country The whole extent was an immense uniform horizon. We were now in the great plain of Lombardy through which the Po rolls his waters into the Adriatic. For many miles beyond Ferrara the soil was marshy, and instead of fences, there were broad ditches covered with water lilies, which served to keep the cows and pigs from straying into the wrong meadows. These were the first swine that I had seen north of the Pontine marshes; and they were poor specimens of the piggish race, lean, scraggy and weak. Wooden barns and stables—another rarity—also struck my sight here. As we proceeded, the aspect of the country became more and more dreary and monotonous. No villages nor even farm-houses were to be seen. Sheds and fences for cattle were all the structures that relieved the universal flatness of the plain. The road became soft and spongy, so that the sound of the wheels and the horses’ feet could no longer be heard. The dikes which extended as far as the eye could reach, like a long green rampart, showed that the country was subject to inundations. In fact, the Po often overflows its banks,
and floods whole miles of the neighboring country In many parts of this alluvial tract are fields of rice, but the cultivation of this article is allowed only to a limited extent, on account of the unwholesome air which the rice plantations are supposed to generate. There were no trees except poplars and willows in long lines on the borders of the ditches: and the most abundant of all living animals seemed to be the frog, which kept up an incessant croaking as we passed.
The sight of the masts of vessels gliding along at a distance beyond an immense line of green wall, apprized us that we were approaching the Po, though we could nowhere discern the current of the river. When we came to that celebrated stream, we found a channel, three quarters of a mile broad, confined on both sides by artificial banks which rise high over the neighboring plains. The waters of the Po are muddy and turbid like the Mississippi, and every three years a great inundation happens in spite of the dikes. On this account, every house is provided with boats, and at the first fall of the heavy rains, which precede the rising of the river, the inhabitants embark with all their goods, so that commonly little damage is caused except to their dwellings, which are of no great value. The river brings down immense quantities of mud, which are deposited at the bottom here, so that the bed of the stream is continually rising.
Crossing the Po by a bridge, we entered the Austrian territory, and submitted to the usual examination from the custom-house officers, after which we were permitted to proceed. The appearance of the country somewhat improved, but still continued equally flat and unvaried in surface. We passed through Rovigo, a decaying and unhealthy town, and travelled over a sandy district to the Adige, which we crossed by a ferry Here were a number of floating mills moored in the stream.
Beyond the Adige, the country becomes hilly, and is diversified with pine and cypress trees. A more cheering prospect appeared in the sight of several neat little villages, full of people, which we were glad to behold, after the journey across the dreary and desolate plain behind us. Still more welcome was the sight of the city of Padua, where we arrived at sunset, and put up for the night.
The streets of Padua are lined with long arcades; it has some pleasant green gardens; the walls of the city are grass-grown, and in taking a walk on the top, I could espy the Alps with their snowy summits away in the north. The general appearance of the city is old fashioned and rusty, and an air of quiet repose reigns in every part. There is a queer-looking old building, which the inhabitants show to travellers, as Livy’s house. I should have wondered if they had not done this, even though the whole city had been levelled to the foundation a dozen times since the day of the great historian. We made only a short stay in Padua, and hurried onward to Venice. A few hours’ travel brought us within sight of the Adriatic, and I beheld the towers of that wonderful city rising out of the waves “as by the stroke of the enchanter’s wand.” It is hardly possible to imagine a more singular spectacle: a great city appears to be floating on the water, and you remain in doubt whether the whole is not an optical illusion.
Embarking in a steamboat, we crossed the lagoon or wide expanse of water which separates the city from the main land. No one has heard of Venice, without forming in his imagination a very distinct and vivid picture of that singular place. Many celebrated cities disappoint expectations, but the preconceptions of Venice are never contradicted or dispelled by the actual view. In a few particulars, the previous notions of the place are not realized, but the general idea is fully verified; and there is besides so much that is strange, unexpected, and magnificent, that the grand impression of the scene is stronger than anticipation. We expect to find it a strange place, and so it proves. The sensation of strangeness, too, remains a long while on the traveller: I have known persons who have lived there for more than a year without getting rid of the feeling of novelty which impressed them on their first arrival. Venice may be characterised as a dreamy place, where a man feels habitually in a sort of transition state between the world he formerly knew, and another one.
On approaching other cities, you hear sounds of life and population, the rattling of wheels, the tramp of footsteps, the cries of the streets. No such sounds here greet the ear. Venice has neither
horses, carriages, nor streets. We shot into a canal between two rows of lofty houses; and though I was prepared by previous descriptions for these long lanes of water, yet the reality of sailing through a city, turning corners and passing under bridges, gave me sensations which I find it impossible to describe. After threading what appeared to me an inextricable maze of these narrow passages, we issued at length into the Grand Canal, which, to compare water with land, may be called the Broadway of Venice. It passes in a curved line through the whole city, and is fronted by splendid marble palaces for almost its whole extent. Across this canal is thrown, in a single arch, the lofty bridge of the Rialto. Black gondolas in great number were gliding up and down the canal, and some small feluccas and river craft lay moored at the quays and landing places.
Palaces, churches and other magnificent structures, the common ornaments of Italian cities, also abound in Venice: but there is no end of describing them. Some of the architecture has a sort of barbarian grandeur, strikingly distinct from that of other parts of this country. There are no monuments of classical times in Venice, for the city grew up after the decline of the Roman Empire. Yet the place is full of objects associated in our memories with historical and poetic recollections. The Place of St. Mark, the Palace of the Doge, suggest stirring remembrances of the celebrated Council of Ten and their mysterious doings. No one can cross the Rialto without thinking of Shylock. The great square of St. Mark is the only place in the city where a man feels as if he were upon terra firma. It is spacious, paved with stone, surrounded with stately buildings, and overhung with lofty towers and domes. Here the crowd of Venice assemble in the evening, sauntering about in the open space, or sipping coffee and sherbet in the shops. The whole place is in a blaze of light from the multitude of coffee houses which line the arcades. The busy hum of chattering crowds, and the lively strains of music which fill the air, render it almost a scene of enchantment, the effect of which is much heightened by its contrast with the stillness and solitude of the remainder of the city. The commerce of Venice brings hither a great crowd of strangers from all parts of the Mediterranean,—Greeks, Turks, Austrians, Dalmatians, Moors, and all the trading population of Europe.
The gondolas of Venice are familiar to every reader They are long, narrow boats, with a prow like the neck of a giraffe, and a house in the middle. In this, you may be rowed all over the city, as you would ride elsewhere shut up in a hackney coach. They are cheap conveyances. You may hire one for a dollar a day, and see all the rarities of the city. For lazy people, I do not know a greater luxury. There is no jolting, nor danger of an overturn, or of horses running away. The gondoliers are civil and obliging, and very useful as guides about the city. They used to sing verses of Tasso by moonlight, but this practice is now discontinued;—the world seems to be growing unpoetical everywhere. It was strange to my eyes, as I rowed down the Grand Canal, under the Rialto, thinking of old times and the golden days and glories of Venice, to come suddenly upon an American brig with “Duan, Boston,” on her stern. Such an apparition amidst the marble palaces of this city of enchantment was the last thing for which I was prepared. She had brought out a cargo of cotton, and was loading with corn for the Boston market.
Gondolas are used not only as hackney coaches, but they also serve the same purpose as handcarts, wheelbarrows, and jackasses in other cities. Country people bring their grain to market, pedlers and hucksters hawk about their commodities, in them. If you want an apple, you hail a gondola. Many of them are rowed by women, and they cry their goods in a demi-musical strain. Most of the traffic of Venice is done in a small way, for though it has some maritime commerce, it is inconsiderable for a city of one hundred and fifty thousand inhabitants. The wealth and population of the place are evidently decaying, and the Austrian government, which prevails in this quarter, is detested by the inhabitants. Many of the palaces are falling to ruin, and the traveller who makes any considerable stay here cannot help seeing proofs that Venice is “dying daily.” Yet, in spite of their misfortunes, the Venetians are cheerful and amiable, and form, what is thought by many, the pleasantest society in Europe. Strange contrast with our own country!—where the people, with every material of enjoyment and prosperity within their reach, in the possession of the most unrestrained political liberty, and amidst the benefits arising from popular education and the general diffusion of knowledge—with a free government, a free press, free commerce,
and unbridled freedom of speech—are so far from being happy or contented, that we pass the greater part of our time in comfortless disquiet, engaged in the barbarous contests of political faction, in unsocial bickering and rivalries, and in the unprofitable indulgence of those sordid and malignant passions which arise from the abuse of the bounties of Providence!
There is one thing which a traveller may learn by visiting foreign lands; namely, that this western world does not enjoy a monopoly of all the wisdom of the universe;—and that, wise and intelligent and enlightened as we are accustomed to call ourselves, it would do no harm to “see ourselves as others see us,” before we attempt to take the first rank among civilized nations. We have yet much to learn and much to do, before we outstrip all other communities in the acquisitions and qualities that render human beings estimable. It is perfectly true, that, as a nation, we possess great advantages, but it is no less true that we have a strong propensity to pervert and abuse them. The good and evil of this world are much more equally balanced than we are apt to suppose. Under tyranny and oppression, the Italians exhibit a cheerfulness and good humor which might put the dissatisfied grumblers of our own land of liberty, to the blush.
Being now compelled to bring my narrative to a close, I shall pass hastily over the remaining portion of my tour. Having satisfied my curiosity with the wonders of Venice, I left that city and proceeded west, along the great plain of Lombardy, to Milan. All the country is cultivated like a garden, and produces the most abundant crops. Everywhere the lofty ridge of the Alps may be seen on the right, like a gigantic wall guarding this rich and beautiful region from the hungry invaders of the North. Yet in all ages this formidable barrier has proved insufficient to preserve Italy from their hostile irruptions. From the times of Brennus and Hannibal, to the present day, army after army of enemies, and swarm after swarm of barbarians have swept over Italy; yet in spite of their ravages, such is the fertility of the soil, such the natural beauties of the territory and the genius of the inhabitants, that it still remains the fairest country upon earth.
Milan is a splendid city, where every house looks like a palace. As the whole city was levelled to the ground about six hundred years ago, there are no marks of antiquity in its buildings. Its enormous cathedral is a gorgeous structure of white marble. The marble statues that cover it on the outside may be counted by thousands. It was begun between four and five hundred years ago, and finished by Napoleon when king of Italy. Milan, however, though full of splendid buildings, does not interest the traveller like most of the other Italian cities. Its level situation takes away everything picturesque from the prospect, and it compares with Florence as Philadelphia does with Boston.
From Milan I proceeded to Turin. This city stands on the Po, and, like very few other places in Italy, appears to be growing fast. The population has rapidly advanced within a few years; and every year adds to the fine structures with which it is embellished. The greater part of the city is regularly built, with straight streets and uniform architecture. There is a very spacious square fronting on the river, with arcades on the three sides. The breadth and open situation of this fine esplanade, with the beautiful prospect of the fresh green hills across the river, render it one of the noblest squares I ever saw. I remarked a handsome bridge just erected across the Dora, a branch of the Po. It is of a stone very similar to Quincy granite, and springs in one wide arch across the stream. The hills around Turin are lofty and picturesque, covered with vineyards, orchards and every kind of fresh verdure. On the top of the highest and steepest of them all, stands the most magnificent church in the whole country Why it was built upon a spot hardly accessible by human feet, one is puzzled to guess, till he learns that it owes its origin to a vow of the Duke of Savoy, previous to a victory obtained over the French in 1706. The church was erected on the spot where the Duke and Prince Eugene stood while they laid the plan of the battle. It is a pity the Duke had not the sense to reflect, that a vow of something useful would have been equally acceptable.
Leaving Turin, I journeyed southeast towards Genoa, crossing, in my course, the plain of Marengo, where the battle was fought which decided the fate of Italy and established the government of Napoleon
over the French. The Alps are in sight at a great distance, and a soldier would say that this wide plain seemed designed for the theatre of a great battle. A steep and rugged road then led me across the Apennines, and the next day I reached Genoa.
I have no space to devote to a description of this noble city, with its hundreds of palaces. I found it necessary to hurry my departure homeward, and took passage in a Genoese felucca for Leghorn, which place I reached in twenty-four hours. Here I was fortunate enough to find a Boston brig on the point of sailing. I embarked in her, and after a long and boisterous passage, landed on Long Wharf, nine months from the day of my departure.
I will add one thing more which may give the reader a notion of the expense of travelling in this quarter of the world. I spent four hundred and fifty dollars on the whole tour, passage out and home included. I visited every considerable Italian city, and resided a reasonable length of time at all the capitals. I lived as well as I could wish, and paid as liberally for everything as any traveller is expected to do. As far as the common objects of travelling are considered, I think there is no other country in which a man can get so much for his money as in Italy.
The Two Friends.
T were once two little boys, who lived near each other in a very pleasant village, near the new forest in Hampshire, England. The name of one was John, and that of the other Paul.
Paul’s father lived in a large house, and kept horses, and servants, and a coach; had a nice lawn and garden, and was, what is called, a gentleman. Paul had a pony to ride on; he had also a great many playthings—tops, hoops, balls, a kite, a ship, and everything he could wish for. He had also fine clothes to wear, and nothing to do but to go to school.
John’s father was a poor man, for he had only a little farm to keep him; and John was forced to get up in the morning and look after the cows, feed the pigs, and do a great deal of work before he went to school.
Although John’s father was a poor man, he was determined to send his son to the best school in the parish: “for,” said he, “if my boy turns out a good lad he will be a comfort to me in my old age.”
When John first came to the school to which Paul went, the boys, who were dressed better than he was, all shunned him. They did not like his rough cord jacket, nor his thick hands and coarse shirt. One said, “he shall not sit by me;” and another said, “he shall not sit by me:” so when he went to a form to sit down, the boy who was on it slid himself to the other end.
Poor John did not know what this meant. At last, when he looked at his coarse clothes, and rough hands, and thick shoes, and compared them with those of his school-fellows, he said to himself, “It is because I am a poor boy:” and the tears came into his eyes.
Paul saw what was going on, and he felt for him, and could have cried too; so he went to the form on which the new scholar sat, and
said, “Do not cry, little boy; I will come and sit by you: here, take this nice rosy apple: do take it; I do not want it! do, there is a dear little boy.”
This made John cry the more; but these were tears of joy, at having found some one to feel for him. He looked at Paul, and sobbed out, “No, no, I thank you.” Then Paul put his arm round his neck, and said, “I cannot bear to see you cry;” and kissed him on the cheek.
One of the boys called out, “Paul Jones is playing with apples;” and, in a minute, the usher came up, and, without making any inquiry, took the apple away, and gave Paul a cut with his cane. The apple he gave to the boy who told, for that was the rule of the school. Paul did not mind the cut, because he knew he was doing right.
Then the other boys laughed, and seemed quite pleased; some peeped from behind their slates, which they held before their faces, as if they were doing their sums; and one called out, in a whisper, “Who likes stick liquorice?”
John felt as if he could have torn the usher to pieces. “Oh!” said he to himself, “if I was a man, see if I would not give it you!” for he felt it cruel that Paul should be struck for being so kind to him.—(It was, however, wrong for him to wish to take revenge.)
From that time, John felt as if he would have died to serve Paul, and he never seemed so happy as when he could play with him, or sit by him at school.
Some time after this happened, Paul, who had about half a mile to walk to his home, through the green lanes, met some gipsies. There were three of them. One said to the other, “Bob, do you see that youngster? He has some good things about him.”
So they whispered a little together. At last, one came close to the little boy, and in a moment seized him round the waist, and put his hand over his mouth and nose, to prevent his calling out. They had made up their minds to steal him for his clothes.
So they put him in a sack, and tied a handkerchief over his mouth, and told him, if he made the least noise they would kill him.
After going for some miles, they went aside into a thick wood; and, when they reached the middle of it, they stripped poor little Paul quite naked, left him under a tree, and went off with all his clothes.
It was now very dark, and Paul was very much frightened. When the gipsies were gone, he cried out for help till he was hoarse, and could cry no longer. Being naked, he was very cold, and he crept under a bush, to screen himself from the wind.
When Paul’s father found he did not come home, he was very unhappy, and went to look for him; he sent servants, first one way, and then another, but no one could find him. His poor mother too was in great grief. Indeed both father and mother were nearly mad through losing him.
They dragged all the ponds in the neighborhood, went up and down the river, inquired of every one they met, but no one had seen him. John was called up, and said, the last time that he saw him was when he bade him good bye, at the corner of the lane.
The night began to close in, and it grew dark; Paul was not found, and poor John was as unhappy as any little boy could be; he went crying to bed, and when he knelt down to say his prayers, he prayed that Paul might come safe home again. He then went to bed, but he could not sleep for thinking of his kind school-fellow.
At last he leaped out of bed, and said, “I must go and see if he is found—I must go and seek him too.” So he slipped on his clothes, let himself out, and fastened the door after him.
At first he did not know what road to take, and he wandered up one lane, and down another. It was very dark at first, so that he could scarcely see where he went. At last the moon rose up, and seemed to cheer him in his search.
So on he walked, looking into every ditch and every pond, going through every little clump of bushes, but to no purpose—he could neither see nor hear anything of poor Paul.
It was about twelve o’clock at night, and he reached the churchyard. Some boys would have been afraid of going into the churchyard, for fear of ghosts. John said to himself, “If the living do not hurt
me, I am sure the dead will not; besides, why should I be afraid, when I am doing what is right.”
John thought he would have one look in the church porch, so he drew towards it. The old arch seemed to frown on him; and it looked so dark within, it made him shudder, although he would not be afraid. He stepped boldly in, and cried, “Paul, are you there?”
Something started with a loud noise, and bounded by him, calling out, “Halloo! halloo!” and leaped to one of the tombstones. When John looked, he found it was a poor silly boy, whom they used to call Silly Mike; and whose part John had often taken, when other boys used to tease him.
“Ah! Mike,” said John, “don’t you know me?” The poor idiot knew him directly, and said, “He is in the sack! he is in the sack!—buried in the wood! Dong, dong—no bell go dong, dong.”
After some trouble, John made Mike understand that he was in search of Paul; who kept saying, he was in a sack in the wood: “Gipsy men,—sack in wood;—Mike frightened.”
At last John prevailed upon the poor fellow to show him to the wood; for the boy thought it might be that Paul had been taken away by somebody.
So they went on till they came to the wood. Mike led the way. At last they thought they heard a moan. John listened:—he heard it again; he then pushed through the brambles, tearing his face and hands at every step.
He called out, “Paul, Paul?” “Here, here,” was faintly said in reply John rushed to the spot, and there lay the poor little boy, half dead.
John ran and helped him up; he then pulled off some of his own clothes, and put them upon him. Mike then lifted him on his back, and they soon got out of the wood.
Paul’s father had been out all night after him. His poor mother had also been searching every place she could think of, and had given him up for lost. They thought he had fallen into the river, and had been drowned.
When the poor lady saw her child borne towards her she could scarcely speak; and, when he leaped into her arms, she fell down in a fainting fit.
Paul’s father soon came home, and was rejoiced to see his son. He took John up also in his arms, and pressed him to his heart, for saving his son.
“I offered a hundred pounds reward to any one who would find him, dead or alive,” said his father. “You shall have the hundred pounds, my little fellow; nay, more, I will give you the best pony in my stable.”
“What for, sir?” said John.
“Why, for being such a brave little fellow.”
“No,” said John, “one good turn deserves another: you remember the nice rosy apple you gave me the first day I went to school, Paul.”
Nothing could prevail upon John, or his father, to take the reward: “To pay my son for doing his duty,” said the poor man, “would spoil all.”
From this time Paul and John were firm friends, and grew up together like brothers. At last Paul became a very rich man, and John was his steward.—English Periodical.
The Selfish Boy.
T selfish boy is one who loves himself solely, and nobody else; who does not care who he deprives of enjoyment, so that he can obtain it. Should he have anything given him, he will keep it all to himself. Should it be a cake, he will keep it in his box, and eat it alone: sometimes creeping up stairs in the day-time, to munch when nobody sees him; at others, getting out of bed at night, to cram himself in the dark.
The selfish boy likes playthings, but he does not like anybody to touch them: “You shall not bowl my hoop; you shall not touch my bat,” is constantly on his tongue. He is ever on the watch, to find out if any one has been even near anything of his. He is restless, anxious, fearful; he knows it lies at the bottom of his heart to rob others, because all selfish boys are covetous, and he thinks that everybody will take from him.
When he sits down to his writing, if he happen to make a good letter he holds his hand over it, so that no one may copy it. When he has worked his sum, he hugs it up to his breast, for fear any one should be benefitted by knowing how it was done; not that it is right to show your sums to others, but this is not his motive.
He obtains knowledge, perhaps works hard for it, but he has no desire for communicating it to others. If he should see a fine sight at the window, he calls for no one to share his delight, but feels a pleasure in being able to say, “I saw it, and you did not.”
The selfish boy cannot see the good of anything, without he is to be the gainer in some way or other. When his interests are concerned, you will see him quite alive, although he was ever so sluggish just before. He sees in a moment what will make to his own advantage, and is, therefore, an adept at chopping and changing, and at making bargains. He knows well enough how to disparage (to
